ceph/src/boost/libs/math/doc/distributions/students_t.qbk

   1 [section:students_t_dist Students t Distribution]
   2
   3 ``#include <boost/math/distributions/students_t.hpp>``
   4
   5    namespace boost{ namespace math{
   6
   7    template <class RealType = double,
   8              class ``__Policy``   = ``__policy_class`` >
   9    class students_t_distribution;
  10
  11    typedef students_t_distribution<> students_t;
  12
  13    template <class RealType, class ``__Policy``>
  14    class students_t_distribution
  15    {
  16       typedef RealType value_type;
  17       typedef Policy   policy_type;
  18
  19       // Construct:
  20       students_t_distribution(const RealType& v);
  21
  22       // Accessor:
  23       RealType degrees_of_freedom()const;
  24
  25       // degrees of freedom estimation:
  26       static RealType find_degrees_of_freedom(
  27          RealType difference_from_mean,
  28          RealType alpha,
  29          RealType beta,
  30          RealType sd,
  31          RealType hint = 100);
  32    };
  33
  34    }} // namespaces
  35
  36 A statistical distribution published by William Gosset in 1908.
  37 His employer, Guinness Breweries, required him to publish under a
  38 pseudonym (possibly to hide that they were using statistics),
  39 so he chose "Student". Given N independent measurements, let
  40
  41 [equation students_t_dist]
  42
  43 where /M/ is the population mean,[' ''' &#x3BC; '''] is the sample mean, and /s/ is the
  44 sample variance.
  45
  46 Student's t-distribution is defined as the distribution of the random
  47 variable t which is  - very loosely - the "best" that we can do not
  48 knowing the true standard deviation of the sample.  It has the PDF:
  49
  50 [equation students_t_ref1]
  51
  52 The Student's t-distribution takes a single parameter: the number of
  53 degrees of freedom of the sample.  When the degrees of freedom is
  54 /one/ then this distribution is the same as the Cauchy-distribution.
  55 As the number of degrees of freedom tends towards infinity, then this
  56 distribution approaches the normal-distribution.  The following graph
  57 illustrates how the PDF varies with the degrees of freedom [nu]:
  58
  59 [graph students_t_pdf]
  60
  61 [h4 Member Functions]
  62
  63    students_t_distribution(const RealType& v);
  64
  65 Constructs a Student's t-distribution with /v/ degrees of freedom.
  66
  67 Requires /v/ > 0, otherwise calls __domain_error.  Note that
  68 non-integral degrees of freedom are supported,
  69 and are meaningful under certain circumstances.
  70
  71    RealType degrees_of_freedom()const;
  72
  73 Returns the number of degrees of freedom of this distribution.
  74
  75    static RealType find_degrees_of_freedom(
  76       RealType difference_from_mean,
  77       RealType alpha,
  78       RealType beta,
  79       RealType sd,
  80       RealType hint = 100);
  81
  82 Returns the number of degrees of freedom required to observe a significant
  83 result in the Student's t test when the mean differs from the "true"
  84 mean by /difference_from_mean/.
  85
  86 [variablelist
  87 [[difference_from_mean][The difference between the true mean and the sample mean
  88                         that we wish to show is significant.]]
  89 [[alpha][The maximum acceptable probability of rejecting the null hypothesis
  90         when it is in fact true.]]
  91 [[beta][The maximum acceptable probability of failing to reject the null hypothesis
  92         when it is in fact false.]]
  93 [[sd][The sample standard deviation.]]
  94 [[hint][A hint for the location to start looking for the result, a good choice for this
  95       would be the sample size of a previous borderline Student's t test.]]
  96 ]
  97
  98 [note
  99 Remember that for a two-sided test, you must divide alpha by two
 100 before calling this function.]
 101
 102 For more information on this function see the
 103 [@http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm
 104 NIST Engineering Statistics Handbook].
 105
 106 [h4 Non-member Accessors]
 107
 108 All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
 109 distributions are supported: __usual_accessors.
 110
 111 The domain of the random variable is \[-[infin], +[infin]\].
 112
 113 [h4 Examples]
 114
 115 Various [link math_toolkit.stat_tut.weg.st_eg worked examples] are available illustrating the use of the Student's t
 116 distribution.
 117
 118 [h4 Accuracy]
 119
 120 The normal distribution is implemented in terms of the
 121 [link math_toolkit.sf_beta.ibeta_function incomplete beta function]
 122 and [link math_toolkit.sf_beta.ibeta_inv_function its inverses],
 123 refer to accuracy data on those functions for more information.
 124
 125 [h4 Implementation]
 126
 127 In the following table /v/ is the degrees of freedom of the distribution,
 128 /t/ is the random variate, /p/ is the probability and /q = 1-p/.
 129
 130 [table
 131 [[Function][Implementation Notes]]
 132 [[pdf][Using the relation: pdf = (v \/ (v + t[super 2]))[super (1+v)\/2 ] / (sqrt(v) * __beta(v\/2, 0.5))   ]]
 133 [[cdf][Using the relations:
 134
 135 p = 1 - z /iff t > 0/
 136
 137 p = z     /otherwise/
 138
 139 where z is given by:
 140
 141 __ibeta(v \/ 2, 0.5, v \/ (v + t[super 2])) \/ 2 ['iff v < 2t[super 2]]
 142
 143 __ibetac(0.5, v \/ 2, t[super 2 ] / (v + t[super 2]) \/ 2   /otherwise/]]
 144 [[cdf complement][Using the relation: q = cdf(-t) ]]
 145 [[quantile][Using the relation: t = sign(p - 0.5) * sqrt(v * y \/ x)
 146
 147 where:
 148
 149 x = __ibeta_inv(v \/ 2, 0.5, 2 * min(p, q))
 150
 151 y = 1 - x
 152
 153 The quantities /x/ and /y/ are both returned by __ibeta_inv
 154 without the subtraction implied above.]]
 155 [[quantile from the complement][Using the relation: t = -quantile(q)]]
 156 [[mode][0]]
 157 [[mean][0]]
 158 [[variance][if (v > 2) v \/ (v - 2) else NaN]]
 159 [[skewness][if (v > 3) 0 else NaN ]]
 160 [[kurtosis][if (v > 4) 3 * (v - 2) \/ (v - 4) else NaN]]
 161 [[kurtosis excess][if (v > 4) 6 \/ (df - 4) else NaN]]
 162 ]
 163
 164 If the moment index /k/ is less than /v/, then the moment is undefined.
 165 Evaluating the moment will throw a __domain_error unless ignored by a policy,
 166 when it will return `std::numeric_limits<>::quiet_NaN();`
 167
 168 (For simplicity, we have not implemented the return of infinity in some cases
 169 as suggested by [@http://en.wikipedia.org/wiki/Student%27s_t-distribution Wikipedia Student's t].
 170 See also [@https://svn.boost.org/trac/boost/ticket/7177].)
 171
 172 [endsect] [/section:students_t_dist Students t]
 173
 174 [/ students_t.qbk
 175   Copyright 2006, 2012 John Maddock and Paul A. Bristow.
 176   Distributed under the Boost Software License, Version 1.0.
 177   (See accompanying file LICENSE_1_0.txt or copy at
 178   http://www.boost.org/LICENSE_1_0.txt).
 179 ]
 180